Computation of Inviscid Incompressible Flow Using the Primitive Variable Formulation

1986 ◽  
Vol 108 (1) ◽  
pp. 68-75 ◽  
Author(s):  
S. Abdallah ◽  
H. G. Smith
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Type Equation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Fine Grid ◽  
Navier Stokes Equations ◽  
Pressure Poisson Equation ◽  
Fine Mesh

The primitive variable formulation originally developed for the incompressible Navier–Stokes equations is applied for the solution of the incompressible Euler equations. The unsteady momentum equation is solved for the velocity field and the continuity equation is satisfied indirectly in a Poisson-type equation for the pressure (divergence of the momentum equation). Solutions for the pressure Poisson equation with derivative boundary conditions exist only if a compatibility condition is satisfied (Green’s theorem). This condition is not automatically satisfied on nonstaggered grids. A new method for the solution of the pressure equation with derivative boundary conditions on a nonstaggered grid [25] is used here for the calculation of the pressure. Three-dimensional solutions for the inviscid rotational flow in a 90 deg curved duct are obtained on a very fine mesh (17 × 17 × 29). The use of a fine grid mesh allows for the accurate prediction of the development of the secondary flow. The computed results are in good agreement with the experimental data of Joy [15].

On Boundary Conditions for Incompressible Navier-Stokes Problems

2006 ◽  
Vol 59 (3) ◽  
pp. 107-125 ◽  
Author(s):  
Dietmar Rempfer
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Review Article ◽  
Navier Stokes ◽  
Field Equations ◽  
Navier Stokes Equations ◽  
Pressure Poisson Equation ◽  
Flow Problems ◽  
Stokes Problems ◽  
Physical Boundary

We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which often do not have immediately obvious “physical” boundary conditions. Most of the issues are discussed for the example of a primitive-variables formulation of the incompressible Navier-Stokes equations in the form of momentum equations plus the pressure Poisson equation. However, analogous problems also exist in other formulations, some of which are briefly reviewed as well. This review article cites 95 references.

scholarly journals Simulation of J-Solution Solving Process of Navier–Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Wenjie Wang ◽  
Melkamu Teshome Ayana
Keyword(s):  
Boundary Conditions ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Simulation Analysis ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Fine Grid ◽  
Navier Stokes Equations ◽  
Advantages And Disadvantages ◽  
Original Boundary

To avoid grid degradation, the numerical analysis of the j-solution of the Navier–Stokes equation has been studied. The Navier–Stokes equations describe the motion of viscous fluid substances. On the basis of the advantages and disadvantages of the Navier–Stokes equations, the incompressible terms and the nonlinear terms are separated, and the original boundary conditions satisfying the j-solution of the Navier–Stokes equation are analyzed. Secondly, the development of a computational grid has been introduced; the turbulence model has also been described. The fluid form and the initial value of the j-solution of the Navier–Stokes equation are combined. The original boundary conditions are solved by a computer, and the nonlinear turbulence equations are derived, which control the fluid flow. The simulation of the fine grid is comprehended to analyze the research outcome. Simulation analysis is carried out to generate multiblock-structured grids with high quality. The j-solution on the grid points is the j-solution that can be used with a fewer number of meshes under the same conditions. The proposed work is easy to implement, and it consumes lesser memory. The results obtained are able to avoid mesh degradation skillfully, and the generated mesh exhibits the characteristics of smoothness, orthogonality, and controllability, which eventually improves the calculation accuracy.

scholarly journals Two-Level Iteration Penalty Methods for the Navier-Stokes Equations with Friction Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Yuan Li ◽  
Rong An
Keyword(s):  
Variational Inequality ◽  
Boundary Conditions ◽  
Variational Inequality Problem ◽  
Stokes Equations ◽  
Mesh Size ◽  
Penalty Methods ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Inequality Problem ◽  
Fine Mesh

This paper presents two-level iteration penalty finite element methods to approximate the solution of the Navier-Stokes equations with friction boundary conditions. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh sizeHin combining with solving a Stokes, Oseen, or linearized Navier-Stokes type variational inequality problem for Stokes, Oseen, or Newton iteration on a fine mesh with mesh sizeh. The error estimate obtained in this paper shows that ifH,h, andεcan be chosen appropriately, then these two-level iteration penalty methods are of the same convergence orders as the usual one-level iteration penalty method.

scholarly journals A SUBGRID STABILIZED METHOD FOR NAVIER-STOKES EQUATIONS WITH NONLINEAR SLIP BOUNDARY CONDITIONS

2021 ◽  
Vol 26 (4) ◽  
pp. 528-547
Author(s):  
Xiaoxia Dai ◽  
Chengwei Zhang
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Fine Mesh ◽  
High Reynolds ◽  
The Stability ◽  
Nonlinear Slip Boundary Conditions

In this paper, we consider a subgrid stabilized Oseen iterative method for the Navier-Stokes equations with nonlinear slip boundary conditions and high Reynolds number. We provide one-level and two-level schemes based on this stability algorithm. The two-level schemes involve solving a subgrid stabilized nonlinear coarse mesh inequality system by applying m Oseen iterations, and a standard one-step Newton linearization problems without stabilization on the fine mesh. We analyze the stability of the proposed algorithm and provide error estimates and parameter scalings. Numerical examples are given to confirm our theoretical findings.

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